不含分部量2的回文有序分拆的一些恒等式

Some Identities for Palindromic Compositions Without $2$'s

中文关键词:  回文有序分拆  Fibonacci数  恒等式  组合证明

英文关键词:palindrome  the Fibonacci number  identity  combinatorial proof

中文摘要:

      本文研究了偶数的互为共轭的分拆都不含分部量2的回文有序分拆,发现这类有序分拆数等于$2F_{n-1}$, 这里 $F_n$表示第$n$个Fibonacc数. 因此,我们得到了几个关于整数的这类回文有序分拆数与分部量是$1, 2$ 的有序分拆数、分部量是奇数的有序分拆数、分部量是大于$1$的有序分拆数之间的一些恒等式.

英文摘要:

      In this paper, we study the palindromic compositions of even integers when no $2$'s are allowed in a composition and its conjugate. We show that the number of these palindromes is equal to $2F_{n-1}$, where, $F_n$ is the $n$-th Fibonacci number. Consequently, we obtain several identities between the number of these palindromes, the number of compositions into parts equal to $1$'s or $2$'s, the number of compositions into odd parts and the number of compositions into parts greater than $1$.